Update AsyncSession documentation and type hints

- Updated default client_identifier from "chrome_120" to "chrome_133"
- Improved type hints using more specific types from the types module
- Rewrote docstrings in reStructuredText format with proper code blocks
- Added more detailed parameter descriptions with examples
- Cleaned up method documentation to be more concise
- Fixed formatting and consistency across the file
- Removed redundant information from docstrings
- Added proper cross-references between parameters
- Improved examples formatting using code blocks

Author: diprog